Filtering in the Spatial Domain Filtering the spatial domain is achieved by convolution Qualitatively: Slide the filter to each position, x, then sum up the function multiplied by the filter at that position

Convolution Example

Convolution Theorem Convolution in the spatial domain is the same as multiplication in the frequency domain –Take a function, f, and compute its Fourier transform, F –Take a filter, g, and compute its Fourier transform, G –Compute H=F  G –Take the inverse Fourier transform of H, to get h –Then h=f  g Multiplication in the spatial domain is the same as convolution in the frequency domain

Filtering Images Work in the discrete spatial domain Convert the filter into a matrix, the filter mask Move the matrix over each point in the image, multiply the entries by the pixels below, then sum –eg 3x3 box filter –averages

filter2(filter,image,shape) filter2(filter,image,'same') is the default; it produces a matrix of equal size to the original image matrix. filter2(filter,image,'valid') applies the mask only to inside pixels. filter2(filter,image,'full') returns a result larger than the original; it does this by padding with zero, and applying the lter at all places on and around the image where the mask intersects the image matrix.

fspecial function; this has many options which makes for easy creation of many different filters. fspecial('average',[5,7]) will return an averaging filter of size 5x7 fspecial('average',11) will return an averaging filter of size 11x11 If we leave out the final number or vector, the 3 x3 averaging filter is returned.

c=imread('cameraman.tif'); f1=fspecial('average'); cf1=filter2(f1,c); figure,imshow(c),figure,imshow(cf1/255)

Using a 9x9 filter Using a 25x25 filter c=imread('cameraman.tif'); f9=fspecial('average', 9); cf9=filter2(f9,c); figure,imshow(c),figure,imshow(cf9/255) f25=fspecial('average', 25); cf25=filter2(f25,c); figure,imshow(c),figure,imshow(cf25/255)

f=fspecial('laplacian') f = >> cf=filter2(f,c); imshow(cf/100)

Laplacian of Gaussian (log) Filter f1=fspecial('log') f1 = >> cf1=filter2(f1,c); figure,imshow(cf1/100) In each case, the sum of all the lter elements is zero.

cf1=filter2(f1,c); figure,imshow(cf1/100) >> f2=[1 -2 1; ;1 -2 1]; cf2=filter2(f2,c); >> figure,imshow(mat2gray(cf2)); mat2gray function automatically scales the matrix elements to displayable values

maxcf2=max(cf2(:)); mincf2=min(cf2(:)); f2g=(cf2-mincf2)/(maxcf2-mincf2); >> imshow(cf2g) >> figure,imshow(cf2/60) We can generally obtain a better result by dividing the result of the filtering by a constant before displaying it:

a=50; s=3; g=fspecial('gaussian', [a a], s); surf(1:a, 1:a, g)

s=9; >> g2=fspecial('gaussian', [a a], s); >> figure, surf(1:a, 1:a, g2)

Edge sharpening

p=imread('pelicans.tif'); u=fspecial('unsharp',0.5); pu=filter2(u,p); imshow(p),figure,imshow(pu/255)

Non-linear Filters maximum filter, which has as its output the maximum value minimum filter, which has as its output the minimum value rank-order filters. In such a filter, the elements under the mask are ordered Colfilt function, which rearranges the image into columns first. median filter, which takes the central value of the ordered list.

geometric mean lter